Devices like digital cameras, film scanners, x-ray machines, and radar imaging devices typically introduce noise into the resulting image. The noise can result from a variety of sources, including sampling errors, temperature-induced “dark current” noise, amplifier noise, and grain in scanned film images. The noise may appear as grain, speckles, and other artifacts. When the amount of noise is sufficiently high, it can detract significantly from the quality of the image, and so it is desirable to remove it.
With many devices, noise is not constant across the image. For instance, well-illuminated areas might have very little noise, while deep shadows may exhibit obvious noise artifacts. The relationship between noise and factors like image brightness is unique for every type of device.
Wavelet thresholding is a technique for removing noise from time-series data, images, and higher-dimensional signals, and it also has application to other areas such as compression. Typically, a wavelet transform is applied to the noisy signal, yielding a set of subbands corresponding to different frequency components of the signal. The noise coefficients in the subbands tend to have smaller magnitude than the coefficients corresponding to edges or other important features. By suppressing these small coefficients, the noise is removed, while the important features are preserved.
The idea of thresholding is not specific to wavelets. For instance, a low-pass filtered version of the original signal can be subtracted from the signal to yield a high-pass filtered version of the signal. Then thresholding can be applied to the high-pass data, and the thresholded high-pass data is added to the low-pass data to reconstruct a denoised signal.
A number of thresholding methods have been proposed in an attempt to improve the quality of thresholding results for wavelet and other noise reduction techniques, including hard thresholding, soft thresholding, sub-band adaptive thresholding, context modeling and frequency thresholding.
In hard thresholding, a coefficient is set to zero if it has absolute value less than a threshold, otherwise it is left unchanged. When used for image denoising, this approach yields undesirable visual artifacts where nearby pixels straddle the threshold value.
Some soft thresholding methods replace a coefficient x by max(0, sgn(x)·(abs(x)−t)), where t is a constant threshold value. This avoids the discontinuity in the hard thresholding function, which yields aesthetically more pleasing results for image denoising.
Hard or soft thresholding can be modified to use a different threshold value t for each subband, where each threshold value is determined by statistical analysis of each subband. In another subband-adaptive approach, Bayesian coring fits a parametric statistical model to each subband in order to exploit kurtosis or other higher-order statistics of the coefficients that can help to distinguish signal from noise. The resulting thresholding function is applied to all coefficients in the subband. That is, the function is not adaptive not within a subband.
In the context modeling approach, a generalized Gaussian distribution is assumed for the coefficients of each wavelet subband. Parameters of the distribution are fit to the coefficients, which requires a potentially time-consuming least-squares estimate of coefficients for a weighted average. The thresholding function is spatially adaptive. In other words, the behavior of the function is modified based on statistical properties of nearby coefficients. The actual threshold value for a particular coefficient is calculated based on a known mathematical relationship between the generalized Gaussian distribution and the noise variance.
The frequency thresholding approach of U.S. Pat. No. 6,741,739 appears to be, at least for conventional wavelet decompositions, algebraically similar to hard thresholding with an infinite threshold value. The method appears to reconstruct a “frequency thresholded” image from the wavelet decomposition using only the high-frequency coefficients of each wavelet level, and setting the low-pass coefficients to zero. This reconstructed image is subtracted from the original image.
Current thresholding methods suffer an additional shortcoming in that they assume that the amount of noise is either signal-independent, or dependent only on monochromatic signal levels in an image. Other methods only relate the noise level to luminance, or assume constant noise throughout a subband.